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GeSi Raman spectra vs. local clustering/anticlustering: Percolation scheme and ab initio calculations
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Image of FIG. 1.
FIG. 1.

Schematic [1 (Ge-Ge), 3 (Ge-Si), 2 (Si-Si)] percolation scheme for random GeSi, as directly inspired from Ref. (refer to Figs. 1 and 4 therein). On the right, the individual oscillators are labeled according to their standard percolation terminology, i.e., using a main term equipped with a superscript and a subscript, in reference to the considered bond-stretching and to the environment in which it takes place, with respect to both composition and length scale, respectively. A numerical labeling is also used (1–6), on the left, for more convenience, notably in a comparison with Figs. 3 and 5 . Limit frequencies calculated in diamond-Si (y ∼ 0, oblique-left hatching) and diamond-Ge (y ∼ 1, oblique-right hatching), on the one hand (open-red squares), and in zincblende-GeSi (y ∼ 0.5, crossed hatching), on the other hand (plain-blue squares), using either pure supercells or containing a unique impurity (as schematically indicated), are used for a qualitative discussion of the frequency shifts of the main GeSi Raman features induced by clustering (red straight-curved arrows) or anticlustering (blue semi-closed loops/arrows). The frequencies of the unique Ge-Ge, main Ge-Si, and Si-Si doublet in random-GeSi, taken from the central curves (  = 0) in Fig. 5 , are added (plain-blue circles), for reference purpose. Globally, the same schematic code and labeling of the limit Raman frequencies is used in Fig. 5 . The -dependence of the individual fractions of oscillators, which monitor directly the Raman intensities, is expressed via the and probabilities in the body of the figure.

Image of FIG. 2.
FIG. 2.

Representative GeSi Raman spectra taken from the literature, used to reveal the effect of local clustering on the (a) Ge-Si (data digitalized from Fig. 1 of Ref. ) and (b) Si-Si (data digitalized from Fig. 1 of Ref. ) fine structures, as emphasized by vertical arrows in each panel. The spectra refer to epitaxial layers grown as random alloys (bottom curves in each panel) or under the form of superlattices (upper curves in each panel), corresponding either to a moderate clustering [  = 0.64, panel (a)] or to a GeSi sequence with interface mixing [panel (b)]. The stars refer to the underlying Si substrate in each case. In panel (a), the upper spectrum is the difference Raman spectrum obtained by subtracting the Raman spectra of the random alloy (bottom curve) from that of the superlattice (  = 0.64), multiplied by 4 (as indicated). Corresponding percolation-type Raman lineshapes (thick-red curves) for the random (bottom curve,  = 0) and clustered (upper curve,  =  0.2) GeSi alloys are superimposed to the experimental data (thin-black ones), for comparison. The horizontal double-arrows mark significant phonon shifts with clustering.

Image of FIG. 3.
FIG. 3.

-dependent percolation (black-thick curves) and MREI (black-thin curves) GeSi Raman lineshapes in case of clustering (  > 0) and anticlustering (  < 0), calculated by using the individual -dependent fractions of oscillators given in Fig. 1 . The Raman frequencies and phonon damping are taken constant, identical to those in the random GeSi alloy, in a crude approximation. The percolation-type Raman spectra corresponding to the selected values of 0.31, 0, and 0.31 are emphasized (red curves). A direct comparison can be made with corresponding -dependent Raman spectra reported in Fig. 5 . Special attention may be awarded to the sensitive Si-Si doublet identified by a specific labeling.

Image of FIG. 4.
FIG. 4.

Positioning of the Si (small-green symbol) and Ge (large-yellow symbol) atoms in three selected 32-atom GeSi supercells corresponding to clustering (  =  0.31, left position) random substitution (  = 0, center position), and anticlustering (  =  0.31, right position). Identical values are obtained just by inverting the Si and Ge atoms in each supercell.

Image of FIG. 5.
FIG. 5.

Raman spectra obtained with the three 32-atom GeSi supercells displayed in Fig. 4 (thick curves), corresponding to local clustering (  =  0.31, top position), random substitution (  = 0, medium position),and local anticlustering (  =  0.31, bottom position). Additional Raman spectra obtained by inverting the positions of the Si and Ge atoms in each supercell (thin curves), thereby leaving the values unchanged, are shifted beneath the original curves, for comparison and identification of intrinsic trends. The frequencies of the unique Ge-Ge, main Ge-Si, and Si-Si doublet in random-GeSi are pointed out (plain-blue circles), for reference purpose. The Ge-Ge, Ge-Si, and Si-Si spectral ranges, delimited by dotted rectangles for help in the discussion, are identified based on limit frequencies when approaching full clustering (top-red arrows, open-red squares) and full anticlustering (bottom-blue arrows, plain-blue squares). Globally, the same schematic code and labeling of such limit Raman frequencies is used as in Fig. 1 . The sensitive Si-Si doublet is emphasized by using numbers (1,2), for unambiguous comparison with the corresponding percolation-type Raman features in Fig. 3 .

Image of FIG. 6.
FIG. 6.

Raman spectra obtained with two 64-atom zincblende GeSi supercells containing either one isolated Si impurity on the Ge sublattice (top spectrum) or one isolated Ge impurity on the Si sublattice (bottom spectrum), as schematically indicated. Distinct modes due to the GeSi-zincblende host matrix and to the isolated Ge and Si impurities are labeled using the same symbol/color code as in Figs. 1 and 5 , for a direct correspondence.


Generic image for table
Table I.

values of the lattice parameter (a), bulk modulus (B), pressure derivative of bulk modulus (B′), and Raman frequency (ω) of pure silicon, pure germanium, and zincblende GeSi, as obtained with different supercell sizes (differentiated by the number of atoms). The k-sampling is specified in each case.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: GeSi Raman spectra vs. local clustering/anticlustering: Percolation scheme and ab initio calculations